Compressibility
The isothermal compressibility of a material is defined as the fractional decrease of volume per unit increase of pressure, at constant temperature. This is the compressibility usually employed in thermodynamic calculations.
Sometimes, as when considering the propagation of sound waves, the adiabatic compressibility is required. This quantity is defined as the fractional decrease of volume per unit increase of pressure, when no heat flows in or out of the system. Because there is no heat flow, the entropy remains unchanged according to the second law of thermodynamics (in a reversible process), so the adiabatic process is also isentropic, i.e., a process that takes place at constant entropy. In adiabatic compression, the temperature rises, so the pressure increases more sharply than in isothermal compression. Therefore, the compressibility at constant entropy is always smaller than that at constant temperature.
It follows from thermodynamic relationships that the ratio of the isentropic (adiabatic) compressibility to the isothermal compressibility is equal to the ratio of the heat capacity at constant volume to the heat capacity at constant pressure.
In the case of an ideal gas, the pressure-volume-temperature relations are governed by the kinetic energy of the gas particles. For denser gases and liquids, it is necessary to take potential energy effects arising from atomic interactions into account, and the pressure-volume-temperature relations tend to be very complex. For crystals, the pressure-volume-temperature relationships depend on the binding forces in the crystal. In ionic crystals such as sodium chloride, the compressibility typically depends on the distance of closest approach of the ions in the crystal under equilibrium conditions, and on the repulsive interactions between ions. In the general case, the calculation of the compressibility of solids reduces to a quantum mechanical problem.
Dilute gases obey the ideal gas law very closely, so the isothermal compressibility is equal to the gas volume divided by the gas constant times the temperature (V/RT); the isentropic compressibility is equal to the isothermal compressibility times the heat capacity at constant volume divided by the heat capacity at constant pressure. However, in the case of compressed gases, the compressibility at high densities falls to a small fraction of the value predicted for the ideal gas.
According to kinetic molecular theory, pressure arises from the impact of gas molecules on container walls. If a gas is compressed at constant temperature, the speed and force of impact of the molecules on the wall remain the same, but the number of collisions per unit area with the container increases. If the gas is compressed adiabatically, however, the heat of compression is not lost, and average molecular speed and force at impact on the walls of the vessel increase as well, so the compressibility is smaller than in the case of isothermal compression.
A related quantity, the bulk modulus, is defined as the reciprocal of the compressibility.
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